The explanation of the first proof

In mathematics, when we restrict ourselves to a finite set of rule and only consider them as legitimate, we will obtain a subset of mathematical object which looks as "a whole" inside them. For example, when we only recognize addition, subtraction, multiplication and division along with natural number as legitimate, what we get is a set of rational number.

There are two kinds of view, one being that

1. The addition, subtraction, multiplication and division of natural numbers compose all the numbers.

2. Hence, object like the square root of 2 is not a number.

The other being that

1. Since the square root of 2 is a number,

2. There is number which can not be obtained from the four operation rules.

There is an inclination to treat the subset of all the object obtained from some legitimate rule as all the things existed, while it is not true.

People may think that in the natural science's view there is a self sustained by the brain functioning and cease to exist when the brain stop functioning. But this is not the orthodox scientific opinion on this matter. While science do not have a dominate view on this matter, a view that could be consider as the most acceptable view in science is that the brain functions, nothing more.

When the brain begin to function, there isn't a self which begin to exist, and in accordance when the brain stop, there isn't a self which cease to exist.

But does it means that the self does not exist. I think the most proper answer to this question is that the self exist but not exist in the world described by natural science. The reason why I consider the self exist can be seen in later section. It is like the square root of 2 is not existed in the set of rational number.

Hence the existence of the self is outside the realm of natural science and is independent of the brain functioning, at least when we only consider the evidence from natural science. Hence when the brain stop functions, there is no need to think that the self cease to exist.

This division of phenomenon and things in themselves is largely Kantian although there maybe different in some detail.